Two Novel Computational Techniques for Solving Nonlinear Time-Fractional Lax’s Korteweg-de Vries Equation

Abstract

This article investigates the seventh-order Lax’s Korteweg–de Vries equation using the Yang transform decomposition method (YTDM) and the homotopy perturbation transform method (HPTM). The physical phenomena that emerge in physics, engineering and chemistry are mathematically expressed by this equation. For instance, the KdV equation was constructed to represent a wide range of physical processes involving the evolution and interaction of nonlinear waves. In the Caputo sense, the fractional derivative is considered. We employed the Yang transform, the Adomian decomposition method and the homotopy perturbation method to obtain the solution to the time-fractional Lax’s Korteweg–de Vries problem. We examined and compared a particular example with the actual result to verify the approaches. By utilizing these methods, we can construct recurrence relations that represent the solution to the problem that is being proposed, and we are then able to present graphical representations that enable us to visually examine all of the results in the proposed case for different fractional order values. Furthermore, the results of the current approach exhibit a good correlation with the precise solution to the problem being studied. Furthermore, the present study offers an example of error analysis. The numerical outcomes obtained by applying the provided approaches demonstrate that the techniques are easy to use and have superior computational performance.

An Efficient Analytical Approaches to Investigate Nonlinear Two-Dimensional Time-Fractional Rosenau-Hyman Equations within the Yang Transform

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On new computations of the time-fractional nonlinear KdV-Burgers equation with exponential memory

“…Notable fractional operators that are used are the Caputo [11,12], Riemann-Liouville [13,14], Caputo-Fabrizo [15], and Atangana-Baleanu [16] operators. These operators are flexible tools for investigating chaotic and fractal events with non-local kernels [17].…”

Novel Analysis of Nonlinear Seventh Order Kaup-Kupershmidt Equation via the Caputo Operator

“…To understand the features of nonlinear problems that arise in everyday life requires an understanding of fractional partial differential equations (FPDEs) numerical and approximate solutions. A variety of mathematical methods that have been created and studied [14,15,16,17,18,19] have been used to obtain the precise solution of NLFDEs. For example, the q-homotopy analysis transform approach for Navier-Stokes equations having fractional-order [20], Natural transform decomposition method for fractional modiőed Boussinesq and approximate long wave equations [24] and fractional-order kaup-kupershmidt equation [23], Yang transform decomposition method for time-fractional Fisher's equation [22] and for time-fractional Noyes-Field model [21], Elzaki homotopy perturbation technique for solving regularized long-wave equations of order fraction [25], Variational iteration transform method for fractional third order Burgers and KdV nonlinear systems [26], Modiőed Khater method for solving nonlinear fractional Ostrovsky equation [27], modiőed ( G ′ G )-expansion scheme for travelling wave solutions of fractional Boussinesq equation [28], generalized Kudryashov method for nonlinear FPDEs of Burgers type [29], Laplace residual power series approach for solving Black-Scholes Option pricing equations having fractional-order [30], őrst integral method for solving fractional Cahn-Allen equation and fractional DSW system [31] and many more [32,33,34,35,36,37,38].…”